Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

TL;DR

This paper investigates the relationship between domination and almost additivity in planar matrix cocycles, extending thermodynamic formalism beyond positive matrices and introducing new conditions for broader applicability.

Contribution

It establishes a characterization of almost additive planar cocycles, linking them to conjugation to isometries or a new weaker domination condition, with applications to matrix thermodynamic formalism.

Findings

Almost additive cocycles are conjugate to isometries or satisfy a weaker domination condition.

Introduces a new property weaker than domination for planar cocycles.

Provides applications to thermodynamic formalism for matrices.

Abstract

In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.